Gaboriau, Damien; Levitt, Gilbert The rank of actions on \(\mathbf R\)-trees. (English) Zbl 0835.20038 Ann. Sci. Éc. Norm. Supér. (4) 28, No. 5, 549-570 (1995). For \(n\geq 2\), let \(F_n\) denote the free group of rank \(n\). We define a total branching index \(i\) for a minimal small action of \(F_n\) on an \(\mathbb{R}\)-tree. We show \(i\leq 2n-2\), with equality if and only if the action is geometric. We thus recover Jiang’s bound \(2n-2\) for the number of orbits of branch points of free \(F_n\)-actions, and we extend it to very small actions (i.e. actions which are limits of free actions).The \(\mathbb{Q}\)-rank of a minimal very small action of \(F_n\) is bounded by \(3n-3\), equality being possible only if the action is free simplicial. There exists a free action of \(F_3\) such that the values of the length function do not lie in any finitely generated subgroup of \(\mathbb{R}\). The boundary of Culler-Vogtmann’s outer space \(Y_n\) has topological dimension \(3n-5\). Reviewer: D.Gaboriau (Lyon) Cited in 3 ReviewsCited in 34 Documents MSC: 20E08 Groups acting on trees 20F65 Geometric group theory 57M07 Topological methods in group theory Keywords:geometric actions; free groups of finite rank; total branching index; minimal small actions; number of orbits of branch points; free \(F_ n\)-actions; small actions; limits of free actions; length functions × Cite Format Result Cite Review PDF Full Text: DOI Numdam EuDML References: [1] R. ALPERIN and H. BASS , Length functions of group actions on \Lambda -trees , in “Combinatorial group theory and topology (S. M. GERSTEN, J. R. STALLINGS, ed.)” (Ann. Math. Studies 111, 1987 , Princeton Univ. Press). MR 89c:20057 | Zbl 0978.20500 · Zbl 0978.20500 [2] M. BESTVINA and M. FEIGHN , Bounding the complexity of simplicial group actions on trees (Inv. Math., Vol. 103, 1991 , pp. 449-469). MR 92c:20044 | Zbl 0724.20019 · Zbl 0724.20019 · doi:10.1007/BF01239522 [3] M. BESTVINA and M. 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